There has been an increasing necessity for scalable optimization methods, especially due to the explosion in the size of data sets and model complexity in modern machine learning applications. Scalable solvers often distribute the computation over a network of processing units. For simple algorithms, such as gradient descent, the dependence of the convergence time with the topology of this network is well known. However, for more involved algorithms, such as the alternating direction method of multipliers (ADMM), much less is known. At the heart of many distributed optimization algorithms, there exists a gossip subroutine which averages local information over the network, whose efficiency is crucial for the overall performance of the method. In this article, we review recent research in this area, and with the goal of isolating such a communication exchange behavior, we compare different algorithms when applied to a canonical distributed averaging consensus problem. We also show interesting connections between ADMM and the lifted Markov chains besides providing an explicit characterization of its convergence and optimal parameter tuning in terms of spectral properties of the network. Finally, we empirically study the connection between network topology and convergence rates for different algorithms on a real-world problem of sensor localization.

Optimization-based models have been used to predict cellular behavior for over 25 years. The constraints in these models are derived from genome annotations, measured macromolecular composition of cells, and by measuring the cell's growth rate and metabolism in different conditions. The cellular goal (the optimization problem that the cell is trying to solve) can be challenging to derive experimentally for many organisms, including human or mammalian cells, which have complex metabolic capabilities and are not well understood. Existing approaches to learning goals from data include (a) estimating a linear objective function, or (b) estimating linear constraints that model complex biochemical reactions and constrain the cell's operation. The latter approach is important because often the known reactions are not enough to explain observations; therefore, there is a need to extend automatically the model complexity by learning new reactions. However, this leads to nonconvex optimization problems, and existing tools cannot scale to realistically large metabolic models. Hence, constraint estimation is still used sparingly despite its benefits for modeling cell metabolism, which is important for developing novel antimicrobials against pathogens, discovering cancer drug targets, and producing value-added chemicals. Here, we develop the first approach to estimating constraint reactions from data that can scale to realistically large metabolic models. Previous tools were used on problems having less than 75 reactions and 60 metabolites, which limits real-life-size applications. We perform extensive experiments using 75 large-scale metabolic network models for different organisms (including bacteria, yeasts, and mammals) and show that our algorithm can recover cellular constraint reactions. The recovered constraints enable accurate prediction of metabolic states in hundreds of growth environments not seen in training data, and we recover useful cellular goals even when some measurements are missing.

The framework of Integral Quadratic Constraints (IQC) introduced by Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov's accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the convergence rate of NAM, and for arbitrary strongly convex functions it was shown numerically that IQC can improve bounds from Nesterov (2004). Unfortunately, an explicit analytic solution to the SDP was not provided. In this paper, we provide such an analytical solution, obtaining a new general and explicit upper bound on the convergence rate of NAM, which we further optimize over its parameters. To the best of our knowledge, this is the best, and explicit, upper bound on the convergence rate of NAM for strongly convex functions.

Several algorithms build on the perfect phylogeny model to infer evolutionary trees. This problem is particularly hard when evolutionary trees are inferred from the fraction of genomes that have mutations in different positions, across different samples. Existing algorithms might do extensive searches over the space of possible trees. At the center of these algorithms is a projection problem that assigns a fitness cost to phylogenetic trees. In order to perform a wide search over the space of the trees, it is critical to solve this projection problem fast. In this paper, we use Moreau's decomposition for proximal operators, and a tree reduction scheme, to develop a new algorithm to compute this projection. Our algorithm terminates with an exact solution in a finite number of steps, and is extremely fast. In particular, it can search over all evolutionary trees with fewer than 11 nodes, a size relevant for several biological problems (more than 2 billion trees) in about 2 hours.

Important data mining problems such as nearest-neighbor search and clustering admit theoretical guarantees when restricted to objects embedded in a metric space. Graphs are ubiquitous, and clustering and classification over graphs arise in diverse areas, including, e.g., image processing and social networks. Unfortunately, popular distance scores used in these applications, that scale over large graphs, are not metrics and thus come with no guarantees. Classic graph distances such as, e.g., the chemical and the CKS distance are arguably natural and intuitive, and are indeed also metrics, but they are intractable: as such, their computation does not scale to large graphs. We define a broad family of graph distances, that includes both the chemical and the CKS distance, and prove that these are all metrics. Crucially, we show that our family includes metrics that are tractable. Moreover, we extend these distances by incorporating auxiliary node attributes, which is important in practice, while maintaining both the metric property and tractability.

The solution path of the one-dimensional fused lasso for an n-dimensional input is piecewise linear with at most O(n) segments. However, existing proofs of this bound do not hold for the weighted fused lasso. At the same time, results for the generalized lasso, of which the weighted fused lasso is a special case, allow more than Ω(3n) segments. In this letter, we prove that the number of segments in the solution path of the weighted fused lasso is O(n2), and that, for some instances, it is Ω(n2). We also give a new, very simple, proof of the O(n) bound for the fused lasso.

The time to converge to the steady state of a finite Markov chain can be greatly reduced by a lifting operation, which creates a new Markov chain on an expanded state space. For a class of quadratic objectives, we show an analogous behavior where a distributed alternating direction method of multipliers (ADMM) algorithm can be seen as a lifting of gradient descent. This provides a deep insight for its faster convergence rate under optimal parameter tuning. We conjecture that this gain is always present, as opposed to the lifting of a Markov chain, which sometimes only provides a marginal speedup.

The framework of Integral Quadratic Constraints of Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). Follow up work by Nishihara et al. (2015) applies this technique to the entire family of over-relaxed Alternating Direction Method of Multipliers (ADMM). Unfortunately, they only provide an explicit error bound for sufficiently large values of some of the parameters of the problem, leaving the computation for the general case as a numerical optimization problem. In this paper we provide an exact analytical solution to this SDP and obtain a general and explicit upper bound on the convergence rate of the entire family of over-relaxed ADMM. Furthermore, we demonstrate that it is not possible to extract from this SDP a general bound better than ours. We end with a few numerical illustrations of our result and a comparison between the convergence rate we obtain for ADMM with known convergence rates for Gradient Descent (GD).

Abstract We present a method to automatically convert videos and CG animations to stylized animated line drawings. Using a data‐driven approach, the animated drawings can follow the sketching style of a specific artist. Given an input video, we first extract edges from the video frames and vectorize them to curves. The curves are matched to strokes from an artist's library, while following the artist's stroke distribution and characteristics. The key challenge in this process is to match the large number of curves in the frames over time, despite topological and geometric changes, allowing to maintain temporal coherence in the output animation. We solve this problem using constrained optimization to build correspondences between tracked points and create smooth sheets over time. These sheets are then replaced with strokes from the artist's database to render the final animation. We evaluate our tracking algorithm on various examples and show stylized animation results based on various artists.

We address the problem of planning collision-free paths for multiple agents using optimization methods known as proximal algorithms. Recently this approach was explored in Bento et al. (2013), which demonstrated its ease of parallelization and decentralization, the speed with which the algorithms generate good quality solutions, and its ability to incorporate different proximal operators, each ensuring that paths satisfy a desired property. Unfortunately, the operators derived only apply to paths in 2D and require that any intermediate waypoints we might want agents to follow be preassigned to specific agents, limiting their range of applicability. In this paper we resolve these limitations. We introduce new operators to deal with agents moving in arbitrary dimensions that are faster to compute than their 2D predecessors and we introduce landmarks, space-time positions that are automatically assigned to the set of agents under different optimality criteria. Finally, we report the performance of the new operators in several numerical experiments.

We describe a new instance-based learning algorithm called the Boundary Forest (BF) algorithm, that can be used for supervised and unsupervised learning. The al- gorithm builds a forest of trees whose nodes store previ- ously seen examples. It can be shown data points one at a time and updates itself incrementally, hence it is nat- urally online. Few instance-based algorithms have this property while being simultaneously fast, which the BF is. This is crucial for applications where one needs to respond to input data in real time. The number of chil- dren of each node is not set beforehand but obtained from the training procedure, which makes the algorithm very flexible with regards to what data manifolds it can learn. We test its generalization performance and speed on a range of benchmark datasets and detail in which settings it outperforms the state of the art. Empirically we find that training time scales as O(DN log(N )) and testing as O(Dlog(N)), where D is the dimensionality and N the amount of data.

Consider the problem of learning the drift coefficient of a p-dimensional stochastic differential equation from a sample path of length T. We assume that the drift is parametrized by a high-dimensional vector, and study the support recovery problem when both p and T can tend to infinity. In particular, we prove a general lower bound on the sample-complexity T by using a characterization of mutual information as a time integral of conditional variance, due to Kadota, Zakai, and Ziv. For linear stochastic differential equations, the drift coefficient is parametrized by a p × p matrix that describes which degrees of freedom interact under the dynamics. In this case, we analyze a ℓ1-regularized least squares estimator and prove an upper bound on T that nearly matches the lower bound on specific classes of sparse matrices.

We describe a novel approach for computing collision-free \emph{global} trajectories for $p$ agents with specified initial and final configurations, based on an improved version of the alternating direction method of multipliers (ADMM) algorithm. Compared with existing methods, our approach is naturally parallelizable and allows for incorporating different cost functionals with only minor adjustments. We apply our method to classical challenging instances and observe that its computational requirements scale well with $p$ for several cost functionals. We also show that a specialization of our algorithm can be used for {\em local} motion planning by solving the problem of joint optimization in velocity space.

We consider the problem of automatically inserting advertisements (ads) into machine composed documents. We explicitly analyze the fundamental tradeoff between expected revenue due to ad insertion and the quality of the corresponding composed documents. We show that the optimal tradeoff a publisher can expect may be expressed as an efficient-frontier in the revenue-quality space. We develop algorithms to compose documents that lie on this optimal tradeoff frontier. These algorithms can automatically choose distributions of ad sizes and ad placement locations to optimize revenue for a given quality or optimize quality for given revenue. Such automation allows a market maker to accept highly personalized content from publishers who have no design or ad inventory management capability and distribute formatted documents to end users with aesthetic ad placement. The ad density/coverage may be controlled by the publisher or the end user on a per document basis by simply sliding along the tradeoff frontier. Business models where ad sales precede (ad-pull) or follow (ad-push) document composition are analyzed from a document engineering perspective.

We present a new paradigm for automated document composition based on a generative, unified probabilistic document model (PDM) that models document composition. The model formally incorporates key design variables such as content pagination, relative arrangement possibilities for page elements and possible page edits. These design choices are modeled jointly as coupled random variables (a Bayesian Network) with uncertainty modeled by their probability distributions. The overall joint probability distribution for the network assigns higher probability to good design choices. Given this model, we show that the general document layout problem can be reduced to probabilistic inference over the Bayesian network. We show that the inference task may be accomplished efficiently, scaling linearly with the content in the best case. We provide a useful specialization of the general model and use it to illustrate the advantages of soft probabilistic encodings over hard one-way constraints in specifying design aesthetics.

Consider the problem of learning the drift coefficient of a stochastic differential equation from a sample path. In this paper, we assume that the drift is parametrized by a high-dimensional vector. We address the question of how long the system needs to be observed in order to learn this vector of parameters. We prove a general lower bound on this time complexity by using a characterization of mutual information as time integral of conditional variance, due to Kadota, Zakai, and Ziv. This general lower bound is applied to specific classes of linear and non-linear stochastic differential equations. In the linear case, the problem under consideration is the one of learning a matrix of interaction coefficients. We evaluate our lower bound for ensembles of sparse and dense random matrices. The resulting estimates match the qualitative behavior of upper bounds achieved by computationally efficient procedures.

We consider the problem of learning a coefficient vector x0 from noisy linear observation y=Ax0+w. In many contexts (ranging from model selection to image processing) it is desirable to construct a sparse estimator. In this case, a popular approach consists in solving an l1-penalized least squares problem known as the LASSO or BPDN. For sequences of matrices A of increasing dimensions, with iid gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic risk of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical models ideas. Through simulations on real data matrices (gene expression data and hospital medical records) we observe that these results can be relevant in a broad array of practical applications.

We consider linear models for stochastic dynamics. Any such model can be associated a network (namely a directed graph) describing which degrees of freedom interact under the dynamics. We tackle the problem of learning such a network from observation of the system trajectory over a time interval T. We analyse the l1-regularized least squares algorithm and, in the setting in which the underlying network is sparse, we prove performance guarantees that are uniform in the sampling rate as long as this is sufficiently high. This result substantiates the notion of a well defined ‘time complexity’ for the network inference problem.